An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

In this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

Total Effective Vascular Compliance of a Global Mathematical Model for the Cardiovascular System

In this work, we determined the total effective vascular compliance of a global closed-loop model for the cardiovascular system by performing an infusion test of 500 mL of blood in four minutes. Our mathematical model includes a network of arteries and veins where blood flow is described by means of a one-dimensional nonlinear hyperbolic PDE system and zero-dimensional models for other cardiovascular compartments. Some mathematical modifications were introduced to better capture the physiology of the infusion test: (1) a physiological distribution of vascular compliance and total blood volume was implemented, (2) a nonlinear representation of venous resistances and compliances was introduced, and (3) main regulatory mechanisms triggered by the infusion test where incorporated into the model. By means of presented in silico experiment, we show that effective total vascular compliance is the result of the interaction between the assigned constant physical vascular compliance and the capacity of the cardiovascular system to adapt to new situations via regulatory mechanisms.

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

In this work, we consider the general family of the so called ADER $$P_NP_M$$ P N P M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step $$P_NP_M$$ P N P M schemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes ($$N=0$$ N = 0 ), the usual Discontinuous Galerkin (DG) methods ($$N=M$$ N = M ), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with $$M>N$$ M > N . In all cases with $$M \ge N > 0 $$ M ≥ N > 0 the $$P_NP_M$$ P N P M schemes are linear in the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of $$P_NP_M$$ P N P M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order $$P_NP_M$$ P N P M schemes, due to the use of a rather fine subgrid of $$2N+1$$ 2 N + 1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.